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IMA Journal of Mathematical Control and Information
, Volume Advance Article (2) – Dec 25, 2016

21 pages

/lp/ou_press/adaptive-full-order-and-reduced-order-observers-for-one-sided-lur-e-XMSlV4bkPV

- Publisher
- Oxford University Press
- Copyright
- © The authors 2016. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
- ISSN
- 0265-0754
- eISSN
- 1471-6887
- D.O.I.
- 10.1093/imamci/dnw066
- Publisher site
- See Article on Publisher Site

Abstract This article proposes an adaptive observer design method for one-sided Lipschitz Lur'e differential inclusion systems with unknown parameters. First, under some assumptions, we design an adaptive full-order observer for the system. Then, under the same assumptions, a reduced-order observer is proved to be valid. Finally, we simulate an example to show the effectiveness of the presented method under the background of rotor system. 1. Introduction With the development of theory and application, many researchers have paid much attention to Lur'e differential inclusion (DI) systems (see Osorio & Moreno 2006; Doris et al. 2008; Bruin et al. 2009; Brogliato & Heemels 2009; Huang et al. 2011a,b). Lur'e DI systems are a kind of nonlinear systems and the most distinguished feature of Lur'e DI systems is that the nonlinear terms contain set-valued mappings. Lur'e DI systems have abroad engineering background, such as circuits systems with ideal diode (see Acary & Brogliato 2008), linear complementary systems (see Brogliato 2003), dynamic systems with Coulomb friction (see Pfeiffer & Hajek 1992, Juloski & Heemels 2004) and so on. Recently, the research of Lur'e DI systems mainly focuses on two aspects. One is the stabilization problem, the aim is to design a controller and make the system absolutely stable or asymptotically stable (see Bruin et al. 2009; Jayawardhana et al. 2009). The other is the observer design problem, based on different conditions of set-valued functions, a lot of different methods have been proposed (see Osorio & Moreno 2006; Doris et al. 2008; Brogliato & Heemels 2009). For the set-valued functions which are upper semi-continuous, closed, convex, bounded and dissipative, Osorio & Moreno (2006) constructed the observer of the Lur'e DI system by dissipative approach. Doris et al. (2008) and Brogliato & Heemels (2009) used positive real method to design the observer for the system. The set-valued functions are upper semi-continuous, closed, convex, bounded and monotone in Doris et al. (2008), while the set-valued functions are not bounded or tight but to be maximal monotone in Brogliato & Heemels (2009). Besides, different parameters in Lur'e DI systems also lead to different methods of observer design (see Huang et al. 2011a, 2013; Zhang et al. 2014). Huang et al. (2011a) considered the Lur'e DI system with unknown parameters, and Zhang et al. (2014) gave further results on the adaptive observer design for the system. Under the framework of stochastic DI, Huang et al. (2013) presented the stochastic observer design method for the Lur'e DI system with Markovian jumping parameters. The recent pioneering work (see Tanwani et al. 2014) dealt with the stability and observer design for the Lur'e DI system, whose set-valued function is not necessarily monotone or time-invariant. The existence of solution as well as the observer design is addressed. As we all know, when the Lipschitz constant becomes large, more results fail to provide a solution. Thus, in order to overcome this drawback, the Lipschitz continuity has been generalized to one-sided Lipschitz continuity. The region of one-sided Lipschitz constant is much larger than that of Lipschitz constant, because the one-sided Lipschitz constant can be zero or even negative while the Lipschitz constant must be positive. This means that one-sided Lipschitz condition is more general than Lipschitz condition, hence one-sided Lipschitz constant which is used to estimate the influence of nonlinear function has been paid extensive close attention (see Hu 2006; Zhao et al. 2010; Abbaszadeh & Marquez 2010; Zhang et al. 2012a,b; Cai et al. 2014). Hu (2006) and Zhao et al. (2010) both considered the observer design problem for nonlinear systems with one-sided Lipschitz term. Hu (2006) proposed an important analysis tool of stability of the error system, and presented sufficient condition of the existence of observer, which was less conservative than the results based on Lipschitz condition in literature. Zhao et al. (2010) improved the result of Hu (2006), and the analysis method is applicable to the system which contains neither monotone nonlinearity nor usual Lipschitz nonlinearity. However, the tractable observer design method is not provided in Hu (2006) or Zhao et al. (2010), because the uncertain constant υp is contained in the constraint conditions. Abbaszadeh & Marquez (2010) investigated one-sided Lipschitz system and presented systematic observer design method, but the existence condition of the observer is relatively strict. After that, further improved results on the former works can be found in Zhang et al. (2012a), Zhang et al. (2012b) and Cai et al. (2014). Zhang et al. (2012a) proposed the observer of one-sided Lipschitz nonlinear system by using the linear matrix inequality (LMI) approach, whereas Zhang et al. (2012b) designed full-order and reduced-order observers for the system via using Riccati equations. Cai et al. (2014) addressed the issue of stabilization and signal tracking control for one-sided Lipschitz DI system. More recently, research results about one-sided Lipschitz nonlinear systems have been presented in Zhang et al. (2015), Huang et al. (2016), Nguyen & Trinh (2016a,b) and Zhang et al. (2016). Huang et al. (2016) considered robust finite-time H∞ control for one-sided Lipschitz system by using the Finsler's lemma, and gave finite-time boundedness conditions for the system. Zhang et al. (2015) studied the nonlinear unknown input observer design problem for one-sided Lipschitz system, which also included well-known Lipschitz counterpart as a special case. Nguyen & Trinh (2016a,b) discussed one-sided Lipschitz time-delay system and one-sided Lipschitz discrete-time system respectively, according to different systems, they proposed different observer design methods. By designing an extended observer structure and using a different Lyapunov function without involving any scaling matrix, Zhang et al. (2016) proposed a unified framework for both full-order and reduced-order exponential state observers. However, to the best knowledge of the authors', the current works all focuses on the differential equation or linear DI, there exists little works on the Lur'e DI with one-sided Lipschitz term. Motivated by above discussion, this article studies observer design method for one-sided Lipschitz Lur'e DI system. The contribution of this article mainly lies in three aspects: (i) We extend the observer design method of nonlinear system to the DI system with one-sided Lipschitz term. Compared to the existing literature, it is a more general DI system. (ii) We present a more simple and effective method to deal with nonlinear terms in nonlinear matrix inequality (NMI). Besides, when dealing with the one-sided Lipschitz term, we derive less conservative and more tractable conditions by using S-procedure. (iii) We prove the existence of the reduced-order observer by construction decomposition method, and the reduced-order observer is more simple than the full-order one. The article is organized as follows: Section 2 presents the problem formulation and some preliminaries. Section 3 designs an adaptive full-order observer for one-sided Lipschitz Lur'e DI system. Section 4 gives the form of reduced-order observer for the same system. Section 5 verifies the effectiveness of the designed observers by numerical simulation. Throughout this article, ||x|| and ||A|| stand for the Euclidean norm of the vector x and the Euclidean norm of the matrix A respectively, xT means the transposition of the vector x, and AT denotes the transposition of the matrix A. rank(A) is the rank of the matrix A, P>(<)0 represents the positive(negative) definite matrix P with P=PT. ⟨⋅,⋅⟩ stands for the inner product, i.e. given x,y∈Rn, then ⟨x,y⟩=xTy. AC means absolutely continuous, while I is the identity matrix. Graph(F) denotes the graph of the set-valued function F(x), i.e. Graph(F)={(x,x*)∣x*∈F(x)}. dom(F) stands for the domain of the set-valued function F(x), i.e. dom(F)={x∣F(x)≠∅}. 2. Problem formulation and preliminaries Consider the following unknown Lur'e DI system {x˙=Ax+Gω+f1(x,u)+Bf2(x,u)θ,ω∈−ρ(Hx),y=Cx, (2.1) where x∈Rn is the state of the system, u∈Rr is the AC control input, and y∈Rq is the measurable output. ρ:Rm→Rm is a set-valued mapping and Hx(0)∈domρ, ω∈Rm is the output of ρ. The constant vector θ∈Rl is unknown, A∈Rn×n, G∈Rn×m, B∈Rn×p, H∈Rm×n, C∈Rq×n are determined matrices. f1(x,u)∈Rn and f2(x,u)∈Rp×l are smooth matrix functions. Without loss of generality, it is assumed that H and C are of full row rank, i.e. rank(H)=m<n and rank(C)=q<n. First, let us recall some basics in Abbaszadeh & Marquez (2010) and Zhang et al. (2012a,b). The nonlinear matrix function f(x,u) is said to be locally Lipschitz in a region D including the origin with respect to x, uniformly in u, ∀x1,x2∈D, if there exists a constant ϑ>0 satisfying ||f(x1,u*)−f(x2,u*)||≤ϑ||x1−x2||, (2.2) where u* is any admissible control signal and the smallest constant ϑ>0 satisfying (2.2) is Lipschitz constant. If condition (2.2) is valid everywhere in Rn, the function f(x,u) is said to be globally Lipschitz. Then, the nonlinear matrix function f(x,u) is said to be one-sided Lipschitz if there exists ζ∈R such that ∀x1,x2∈D ⟨f(x1,u*)−f(x2,u*),x1−x2⟩≤ζ||x1−x2||2, (2.3) where ζ∈R is called the one-sided Lipschitz constant. Actually, it is obvious that any Lipschitz function is also one-sided Lipschitz. Finally, the nonlinear matrix functions f(x,u) is called quadratic inner-boundedness in the region D˜ if ∀x1,x2∈D˜, there exist β,γ∈R such that (f(x1,u)−f(x2,u))T(f(x1,u)−f(x2,u))≤β||x1−x2||2+γ⟨x1−x2,f(x1,u)−f(x2,u)⟩. (2.4) The Lipschitz function is also quadratically inner-bounded with γ=0 and β>0. Thus, the Lipschitz continuity implies quadratic inner-boundedness. However, the converse is not true. Figure 1 shows the relation between the Lipschitz, one-sided Lipschitz and quadratically inner-bounded function sets. Fig. 1. View largeDownload slide The Lipschitz, one-sided Lipschitz and quadratically inner-bounded function sets. Fig. 1. View largeDownload slide The Lipschitz, one-sided Lipschitz and quadratically inner-bounded function sets. We now introduce some definitions about DI, which can be referred to Smirnov (2002). Definition 2.1 (see Smirnov (2002)) Let F:Rn→Rn be a given set-valued mapping. If the function x:[0,T]→Rn is AC for each x0∈Rn and satisfies x˙(t)∈F(x(t)) for almost all t∈[0,T], then x(t) is called a solution to the DI x˙(t)∈F(x(t)), t∈[0,T] with x(0)=x0∈domF. Definition 2.2 (see Smirnov 2002) Let G:Rn→Rn be a given set-valued mapping. If the following holds (y−y*)T(x−x*)≥0, ∀(x,y),(x*,y*)∈Graph(G), (2.5) then G(x) is monotone. In order to obtain the main results, the following assumptions are needed. Assumption 1 The set-valued mapping ρ(⋅) satisfies the following properties: Assumption 1-1 ρ(⋅) is upper semi-continuous, non-empty, convex, closed and bounded. Assumption 1-2 ρ(⋅) is monotone. Assumption 1-3 there exist positive constants c1 and c2 such that for any ω∈−ρ(z) ||ω||≤c1||z||+c2. Assumption 2 f1(x,u) is one-sided Lipschitz in a region D1 and quadratically inner-bounded in a region D˜1, i.e. ∀x,x^∈D1∩D˜1, there exist ζ,β,γ∈R such that ⟨f1(x,u)−f1(x^,u),x−x^⟩≤ζ||x−x^||2, (2.6) (f1(x,u)−f1(x^,u))T(f1(x,u)−f1(x^,u))≤β||x−x^||2+γ⟨x−x^,f1(x,u)−f1(x^,u)⟩. (2.7) f2(x,u) is Lipschitz in a region D¯2, i.e. ∀x,x^∈D¯2, there exists ϑ>0 such that ||f2(x,u)−f2(x^,u)||≤ϑ||x−x^||. (2.8) Assumption 3 The unknown parameter vector θ is bounded with μ>0, i.e. ||θ||≤μ. (2.9) Assumption 4 Let α=ϑμ||B||, where ϑ and μ are defined in (2.8) and (2.9). There exist constants ε1>0, ε2>0, and ε>0, matrices P∈Rn×n>0, L∈Rn×q, F∈Rm×q, M∈Rp×q such that P(A−LC)+(A−LC)TP+αP2+(α+ε+ε1ζ+ε2β)I+1ε2(P+γε2−ε12I)2≤0, (2.10) GTP=H−FC, (2.11) BTP=MC. (2.12) Some useful lemmas are given as follows, which are necessary for our study. Lemma 2.1 (see Smirnov 2002) Let F be a set-valued function, we assume that F is upper semicontinuous, closed, convex and bounded for all x∈Rn. Then, for each x0∈Rn there exists an AC function x(t) defined on [0,T], which is a solution of the initial value problem x˙(t)∈F(x(t)), t∈[0,T] with x(0)=x0∈domF. By Edmond & Thibault (2005) and Tanwani (2014), the following lemma is a natural extension of Lemma 2.1. Lemma 2.2 If F is upper semicontinuous, closed, convex and bounded for all x∈Rn, then the solution of x˙∈F(x)+f(x,u) defined on [0,T] exists, where f(x,u) is a one-sided Lipschitz function and x(0)=x0∈domF. Lemma 2.3 (see Krstic & Deng 1998) Let φ:R+→R be a known function. If φ is uniformly continuous and the integral ∫0∞φ(s)ds exists, then limt→∞φ(t)=0. Lemma 2.4 (see Zhang et al. 2012b) For a given matrix S=[S11S12S12TS22] with S11T=S11 and S22T=S22, the following conditions are equivalent: (1) S<0, (2) S11<0,S22−S12TS11−1S12<0, (3) S22<0,S11−S12S22−1S12T<0. 3. Adaptive full-order observer The adaptive full-order observer for the system (2.1) is designed as {x^˙=Ax^+Gω^+f1(x^,u)+Bf2(x^,u)θ^+L(y−Cx^),ω^∈−ρ(Hx^+F(y−Cx^)),y=Cx^, (3.1) with θ^˙=f2T(x^,u)M(y−Cx^), (3.2) where (H−FC)x^(0)+Fy(0)∈domρ. Remark 3.1 By a fundamental conclusion in Kreyszig (1978), boundedness of a linear mapping is equivalent to its continuity, thus Ax and Ax^ are both bounded. Thus, the mappings (t,x)→Ax−Gρ(Hx)+f1(x,u)+Bf2(x,u)θ and (t,x)→Ax^−Gρ(Hx^+F(y−Cx^))+f1(x^,u)+Bf2(x^,u)θ^+L(y−Cx^) are upper semi-continuous, non-empty, closed, convex, and bounded. In view of Lemma 2.2, Assumption 1-1 guarantees the local existence of the solutions of the system (2.1) and the observer (3.1). Due to Assumption 1-3 (known as the growth condition), finite escape times can be prevented and thus solutions to the system (2.1) and the observer (3.1) are globally defined on [0,+∞). Denote that e=x−x^, θ˜=θ−θ^ and f˜i=fi(x,u)−fi(x^,u),i=1,2. By (2.1) and (3.1), we have the following error system {e˙=(A−LC)e+G(ω−ω^)+f1˜+Bf2˜θ+Bf2(x^,u)θ˜,ω∈−ρ(Hx),ω^∈−ρ(Hx^+F(y−Cx^)). (3.3) Based on the error system (3.3), we can state the theorem as follows. Theorem 3.1 If Assumptions 1–4 hold, (3.1) is an adaptive full-order observer of the system (2.1), i.e. e(t) of the error system (3.3) satisfies limt→∞e(t)=0. Proof. Consider the following Lyapunov candidate: V=eTPe+θ˜Tθ˜. (3.4) Along the trajectories of the error system (3.3), the derivative of V can be calculated as: V˙=2eTPe˙+2θ˜Tθ˜˙=2eTP[(A−LC)e+G(ω−ω^)+f1˜+Bf2˜θ+Bf2(x^,u)θ˜]+2θ˜Tθ˜˙=2eTP(A−LC)e+2eTPG(ω−ω^)+2eTPf1˜+2eTPBf2˜θ+2eTPBf2(x^,u)θ˜+2θ˜Tθ˜˙. (3.5) By Assumption 3 and (2.8), we have 2eTPBf2˜θ≤2||eTP||||B||||f˜2||||θ||≤2ϑμ||B||||eTP||||e||≤α(eTP2e+eTe). (3.6) From (2.12) and (3.2), we can obtain that 2eTPBf2(x^,u)θ˜+2θ˜Tθ˜˙=2eTCTMTf2(x^,u)θ˜−2θ˜Tf2T(x^,u)M(y−Cx^)=0. (3.7) By (2.11) and the monotonicity of ρ(⋅), we have 2eTPG(ω−ω^)=2eT(H−FC)T(ω−ω^)=−2[Hx−((H−FC)x^+Fy)]T[−ω−(−ω^)]≤0. (3.8) Substituting (3.6)–(3.8) into (3.5) yields V˙≤eT[P(A−LC)+(A−LC)TP+αP2+αI]e+2eTPf˜1=[ef˜1]T[P(A−LC)+(A−LC)TP+αP2+αIPP0][ef˜1]. (3.9) From (2.6), we get ζeTe−eTf˜1≥0. Therefore, the following holds ε1[ef˜1]T[ζI−I2∗0][ef˜1]≥0. (3.10) Similarly, in view of (2.7), we have ε2[ef˜1]T[βIγ2I∗−I][ef˜1]≥0. (3.11) Then, combining (3.10) and (3.11) with (3.9) yields V˙≤[ef˜1]T[P(A−LC)+(A−LC)TP+αP2+(α+ε1ζ+ε2β)IP+γε2−ε12I∗−ε2I][ef˜1]. (3.12) Using Lemma 2.4, (2.10) is equivalent to [P(A−LC)+(A−LC)TP+αP2+(α+ε+ε1ζ+ε2β)IP+γε2−ε12I∗−ε2I] =[P(A−LC)+(A−LC)TP+αP2+(α+ε1ζ+ε2β)IP+γε2−ε12I∗−ε2I]+[εI000]≤0. (3.13) Substituting (3.13) into (3.12) results in V˙≤−εeTe. (3.14) Integrating both sides from 0 to t yields V(t)≤V(0)−∫0tεeT(s)e(s)ds. (3.15) Since V>0, (3.15) becomes ∫0tεeT(s)e(s)ds≤V(0), which deduces to limt→∞∫0tεeT(s)e(s)ds≤V(0)<∞. (3.16) By Lemma 2.3, we obtain limt→∞εeT(t)e(t)=0, (3.17) which means limt→∞e(t)=0. (3.18) Thus, the proof is completed. □ Remark 3.2 The connection with spectrum of A−LC with the one-sided Lipschitz constant has been given in Abbaszadeh & Marquez (2010). Similarly, (2.10) can also be written as (A−LC)TP+P(A−LC)+αP2+αI≤−Q,ξλmax(P)−λmin(P)<α¯λmin(Q),γ+2α¯>0,λmax(P)λmin(P)(α¯2−1)≤α¯2, where α¯>0 and ξ=(β+1)+ζ(γ+α¯2). It should be noted that we obtain the main results by S-procedure in this article, and the feasible solutions of (2.10)–(2.12) can be solved. Remark 3.3 Let Y=PL, (2.10) can be written as P(A−LC)+(A−LC)TP+αP2+(α+ε+ε1ζ+ε2β)I+1ε2(P+γε2−ε12I)2 =PA+ATP−YC−CTYT+(α+ε+ε1ζ+ε2β+(γε2−ε1)24ε2)I +γε2−ε1ε2P+(α+1ε2)P2≤0, (3.19) which is equivalent to [PA+ATP−YC−CTYT+(α+ε+ε1ζ+ε2β+(γε2−ε1)24ε2)I+γε2−ε1ε2Pα+1ε2Pα+1ε2P−I]≤0. (3.20) Then (3.20) is a LMI while (2.11)-(2.12) are linear matrix equalities (LMEs), and we can solve the solutions of P, Y and L by Scilab. Remark 3.4 It should be noted that Tanwani et al. (2014) considered the Lur'e DI system whose set-valued mapping is described by a general form, i.e. a kind of normal cone. The model studied in this article can be included by Tanwani et al. (2014) because the normal cone operator associated with certain class of non-convex sets satisfies the one-sided Lipschitz inequality. However, the model studied in this article is different from that in Hu (2006) and Zhao et al. (2010). Hu (2006) and Zhao et al. (2010) both focused on nonlinear systems with one-sided term, we studied Lur'e DI system which contains one-sided nonlinear term and uncertain parameters in this article. Besides, the method used in this article is different from that in Hu (2006), Zhao et al. (2010) and Tanwani et al. (2014). Hu (2006) and Zhao et al. (2010) mainly employed the matrix theory and Lyapunov stability to deal with the convergence of the error system, Tanwani et al. (2014) used the mathematical analysis and DI theory to give the strict existence condition of solution as well as the stability. We utilize the S-procedure and LMI method to deal with the one-sided Lipschitz term, and then employ the adaptation technique (such as (3.2) to design the adaptive observer. 4. Reduced-order observer Under the same conditions as that in Section 3, we prove the existence of the reduced-order observer for the system (2.1) in this section. Without loss of generality, we suppose that C=[Iq0], where Iq is the identity matrix. Meanwhile, we decompose x, A, G, B, H, P, f1(x,u) into the forms as follows: x=[x1x2], A=[A11A12A21A22], G=[G1G2], B=[B1B2],H=[H1H2], P=[P11P12P12TP22], f1(x,u)=[f11(x,u)f12(x,u)], where x1∈Rq, A11∈Rq×q, G1∈Rq×m, B1∈Rq×p, H1∈Rm×q, P11∈Rq×q, f11(x,u)∈Rq, then the system (2.1) can be written as {x˙1=A11x1+A12x2+G1ω+f11([x1x2],u)+B1f2([x1x2],u)θ,x˙2=A21x1+A22x2+G2ω+f12([x1x2],u)+B2f2([x1x2],u)θ,ω∈−ρ(H1x1+H2x2),y=x1. (4.1) Substituting y=x1 into (4.1) yields {y˙=A11y+A12x2+G1ω+f11([yx2],u)+B1f2([yx2],u)θ,x˙2=A21y+A22x2+G2ω+f12([yx2],u)+B2f2([yx2],u)θ,ω∈−ρ(H1y+H2x2). (4.2) Let z=x2−Ky and K∈R(n−q)×q, then (4.2) can be transformed into {z˙=(A22−KA12)z+(G2−KG1)ω+[(A21−KA11)+(A22−KA12)K]y+Φ1+Φ2,ω∈−ρ(H2z+(H1+H2K)y),x2=z+Ky, (4.3) where Φ1=[−KIn−q]f1([yz+Ky],u), (4.4) Φ2=[−KIn−q]Bf2([yz+Ky],u)θ. (4.5) Theorem 4.1 Let K=−P22−1P12T and Ψ=P22(A22−KA12)+(A22−KA12)TP22, then Ψ+(ε+ε1ζ+ε2β)In−q+P22KKTP22ε2+1ε2(P22+γε2−ε12In−q)2≤0, (4.6) (G2−KG1)TP22=H2, (4.7) [−KIn−q]B=0. (4.8) Proof. Since C=[Iq0], the block form of (2.10) is [**∗P12TA12+A12TP12+P22A22+A22TP22+Γ]≤0, (4.9) where Γ=α(P12TP12+P222+In−q)+(ε+ε1ζ+ε2β)In−q+P12TP12ε2+1ε2(P22+γε2−ε12In−q)2. (4.10) By Lemma 2.4, the following holds P12TA12+A12TP12+P22A22+A22TP22+Γ≤0. (4.11) It is obvious that the first term of Γ is positive definite, that is α(P12TP12+P222+In−q)>0. Substituting K=−P22−1P12T into (4.6), we have Ψ+(ε+ε1ζ+ε2β)In−q+P22KKTP22ε2+1ε2(P22+γε2−ε12In−q)2 =P12TA12+A12TP12+P22A22+A22TP22+(ε+ε1ζ+ε2β)In−q+P12TP12ε2 +1ε2(P22+γε2−ε12In−q)2≤P12TA12+A12TP12+P22A22+A22TP22+Γ≤0. (4.12) The block form of (2.11) is [∗G1TP12+G2TP22]=[∗H2], (4.13) then G1TP12+G2TP22=H2, (4.14) (4.7) can be computed by (G2−KG1)TP22=(G2+P22−1P12TG1)TP22=G1TP12+G2TP22=H2. (4.15) Similarly, the block form of (2.12) is BT[∗P12∗P22]=M[∗0], (4.16) which implies that BT[P12P22]=0, (4.17) i.e. [P12TP22]B=0. (4.18) Thus [−KIn−q]B=[P22−1P12TIn−q]B=P22−1[P12TP22]B=0. (4.19) □ Then, we propose the reduced-order observer for the system (2.1), which has the following form: {z^˙=(A22−KA12)z^+(G2−KG1)ω^+[(A21−KA11)+(A22−KA12)K]y+Φ^1,ω^∈−ρ(H2z^+(H1+H2K)y),x^2=z^+Ky, (4.20) where Φ^1=[−KIn−q]f1([yz^+Ky],u). (4.21) Theorem 4.2 Let K=−P22−1P12T. If Assumptions 1–4 hold, then (4.20) is a reduced-order observer of the system (2.1). i.e. limt→∞[x2(t)−x^2(t)]=0. Proof. By (4.8), we have Φ2=0, then (4.3) can be written as {z˙=(A22−KA12)z+(G2−KG1)ω+[(A21−KA11)+(A22−KA12)K]y+Φ1,ω∈−ρ(H2z+(H1+H2K)y),x2=z+Ky. (4.22) Let ez=z−z^ and Φ˜1=Φ1−Φ^1, subtracting (4.20) from (4.22), we can obtain the following error system {e˙z=(A22−KA12)ez+(G2−KG1)(ω−ω^)+Φ˜1,ω∈−ρ(H2z+(H1+H2K)y),ω^∈−ρ(H2z^+(H1+H2K)y). (4.23) Choose the Lyapunov candidate as V(ez)=ezTP22ez. Taking the derivative of V(ez) along the trajectories of (4.23) results in V˙(ez)=2ezTP22e˙z=ezT[P22(A22−KA12)+(A22−KA12)TP22]ez+2ezTP22(G2−KG1)(ω−ω^)+2ezTP22Φ˜1=ezTΨez+2ezTP22(G2−KG1)(ω−ω^)+2ezTP22Φ˜1. (4.24) By (4.7) and the monotonicity of ρ(⋅), we have 2ezTP22(G2−KG1)(ω−ω^)=2ezTH2T(ω−ω^) =−2[(H2z+(H1+H2K)y)−(H2z^+(H1+H2K)y)]T[−ω−(−ω^)]≤0. (4.25) Then we can obtain that 2ezTP22Φ˜1=2ezTP22[−KIn−q]f˜1=2ezT[P12TP22][f˜11f˜12], (4.26) where f˜1i=f1i([yz+Ky],u)−f1i([yz^+Ky] ,u) ,i=1,2. Given (4.24)–(4.26), the following expression holds V˙(ez)≤ezTΨez+2ezT[P12TP22][f˜11f˜12]=[ezf˜11f˜12]T[ΨP12TP22P1200P2200][ezf˜11f˜12] . (4.27) Using the one-sided Lipschitz condition (2.6), we have ⟨f˜1,[0ez]⟩≤ζ||[0ez]||2. (4.28) The above inequality implies that f˜12Tez≤ζezTez. Therefore, ε1[ezf˜11f˜12]T[ζIn−q0−In−q2∗00∗∗0][ezf˜11f˜12]≥0. (4.29) Similarly, it follows from the condition (2.7) of quadratic inner-boundedness that f˜1Tf˜1≤β||[0ez]||2+γ⟨[0ez],f˜1⟩, (4.30) which means that f˜11Tf˜11+f˜12Tf˜12≤βezTez+γezTf˜12. (4.31) Thus, ε2[ezf˜11f˜12]T[βIn−q0γIn−q2∗−Iq0∗∗−In−q][ezf˜11f˜12]≥0. (4.32) Then, adding the left terms of (4.29) and (4.32) to the right-hand side of (4.27) yields V˙(ez)≤[ezf˜11f˜12]TΛ[ezf˜11f˜12] , (4.33) where Λ=[Ψ+(ε1ζ+ε2β)In−qP12TP22+γε2−ε12In−qP12−ε2Iq0P22+γε2−ε12In−q0−ε2In−q]. (4.34) Using Lemma 2.4, (4.6) is equivalent to Λ+[εIn−q00000000]<0, then it follows from (4.33) that V˙(ez)≤−εezTez. Therefore, we can conclude that limt→∞[z^(t)−z(t)]=0, i.e. limt→∞[x^2(t)−x2(t)]=0. Thus, we complete the proof. □ Remark 4.1 Compared with adaptive full-order observer (3.1), the reduced-order observer (4.20) has advantage because it has lower dimension than full-order one. Besides, the reduced-order observer does not contain any adaption law. 5. Numerical example In this section, we take the background of rotor system (see Doris et al. 2008). The rotor system with friction is an experimental set-up consisting of upper and lower steel disc, which are connected through a low stiffness steel string. The upper disc is drived by DC-motor with bounded input voltage. Both discs can rotate around their respective geometric centres and the related angular positions are measured by incremental encoders. Then, we consider the following Lur'e DI system with unknown parameters: {x˙=Ax+Gω+f1(x,u)+Bf2(x,u)θ,ω∈−ρ(Hx),y=Cx, (5.1) where A=[01−1−0.1526−4.668802.230100.6442] , G=[0030.6748] , B=[221] ,f1(x,u)=[2u−x2(x22+x32)+u−x3(x22+x32)+u] , f2(x,u)=0.1cosx3, H=[001], C=[100]. By Doris et al. (2008), the set-valued mapping ρ(κ) is defined by ρ(κ)={[0.1642+0.0603(1−21+e5.7468|κ|)−0.2267(1−21+e0.2941|κ|)]sign(κ)+0.0319κ,κ≠0,[−0.1642,0.1642],κ=0. From Abbaszadeh & Marquez (2010) or Zhang et al. (2012a,b), we know that f1(x,u) is globally one-sided Lipschitz with one-sided Lipschitz constant ζ=0. In what follows, we focus on the set D˜1={x∈R3:||x||≤r}. Let r=min(−γ4,β+γ244) , γ<0, β+γ24>0. Then one can verify the quadratically inner-bounded property of f1(x,u) in D˜1. f1(x,u) is globally one-sided Lipschitz, that is D1=R3, D1∩D˜1=D˜1. Note that the region D˜1 can be made arbitrarily large by choosing appropriate values for γ and β. In the meantime, f2(x,u) is globally Lipschitz with the Lipschitz constant ϑ=0.1, ||B||=3. Let the unknown parameter θ=0.1, then we can obtain that μ=0.1 and α=ϑμ||B||=0.03. Based on the above results, we take β=−399, γ=−40 and solve (2.10)–(2.12) by Scilab, then P=[0.0902−0.0248−0.0163−0.02480.02480−0.016300.0326] , L=[11.522158.360817.2868] , F=0.5000,M=0.1140, ε=0.0082, ε1=0.0148, ε2=0.0254. Thus, the adaptive full-order observer is designed as follows: {x^˙1=−11.5221x^1+x^2−x^3+0.2cosx^3θ^+11.5221y+2u,x^˙2=−58.5134x^1−4.6688x^2−x^2(x^22+x^32)+0.2cosx^3θ^+58.3608y+u,x^˙3=−15.0567x^1+0.6442x^3−x^3(x^22+x^32)+0.1cosx^3θ^+30.6748ω^−17.2868y+u,ω^∈−ρ(−0.5x^1+x^3+0.5y),θ^˙=0.1140cosx^3(y−x^1). (5.2) The reduced-order observer has the following form: {z^˙1=−5.6688z^1+z^2−5.3214y−(z^1+y)[(z^1+y)2+(z^2+0.5y)2]−u,z^˙2=−0.5z^1+1.1442z^2+30.6748ω^+2.3022y−(z^2+0.5y)[(z^1+y)2+(z^2+0.5y)2],ω^∈−ρ(z^2+0.5y),x^2=z^1+y,x^3=z^2+0.5y. (5.3) Similarly, let θ=1, then we obtain P=[0.2681−0.0611−0.0163−0.06110.06110−0.016300.0326], L=[40.960311.205858.4020], F=0.5000,M=0.3980, ε=0.0043, ε1=0.0096, ε2=0.0073. Thus, the adaptive full-order observer is designed as follows: {x^˙1=−40.9603x^1+x^2−x^3+0.2cosx^3θ^+40.9603y+2u,x^˙2=−11.3584x^1−4.6688x^2−x^2(x^22+x^32)+0.2cosx^3θ^+11.2058y+u,x^˙3=−56.1719x^1+0.6442x^3−x^3(x^22+x^32)+0.1cosx^3θ^+30.6748ω^+58.4020y+u,ω^∈−ρ(−0.5x^1+x^3+0.5y),θ^˙=0.3980cosx^3(y−x^1). (5.4) The reduced-order observer has the following form: {z^˙1=−5.6688z^1+z^2−5.3214y−(z^1+y)[(z^1+y)2+(z^2+0.5y)2]−u,z^˙2=−0.5z^1+1.1442z^2+30.6748ω^+2.3022y−(z^2+0.5y)[(z^1+y)2+(z^2+0.5y)2],ω^∈−ρ(z^2+0.5y),x^2=z^1+y,x^3=z^2+0.5y. (5.5) We now complete the simulation by using the simulink in Matlab. When θ=0.1, as shown in Figs 2–4, the estimated state trajectories of the adaptive full-order observer (5.2) converge to the state trajectories of the system (5.1). Figures 5 and 6 show that the estimated state trajectories of the reduced-order observer (5.3) converge to the state trajectories of the system (5.1). Similarly, when θ=1, as shown in Figs 7–9, the estimated state trajectories of the adaptive full-order observer (5.4) converge to the state trajectories of the system (5.1). Figures 10 and 11 show that the estimated state trajectories of the reduced-order observer (5.5) converge to the state trajectories of the system (5.1). In view of the simulation results, the observer design method is both effective when θ=0.1 and θ=1. Thus, the proposed design method is effective in this article. Fig. 2. View largeDownload slide The state x1 of system (5.1) and the estimated state x^1 of adaptive full-order observer (5.2) when θ=0.1. Fig. 2. View largeDownload slide The state x1 of system (5.1) and the estimated state x^1 of adaptive full-order observer (5.2) when θ=0.1. Fig. 3. View largeDownload slide The state x2 of system (5.1) and the estimated state x^2 of adaptive full-order observer (5.2) when θ=0.1. Fig. 3. View largeDownload slide The state x2 of system (5.1) and the estimated state x^2 of adaptive full-order observer (5.2) when θ=0.1. Fig. 4. View largeDownload slide The state x3 of system (5.1) and the estimated state x^3 of adaptive full-order observer (5.2) when θ=0.1. Fig. 4. View largeDownload slide The state x3 of system (5.1) and the estimated state x^3 of adaptive full-order observer (5.2) when θ=0.1. Fig. 5. View largeDownload slide The state x2 of system (5.1) and the estimated state x^2 of reduced-order observer (5.3) when θ=0.1. Fig. 5. View largeDownload slide The state x2 of system (5.1) and the estimated state x^2 of reduced-order observer (5.3) when θ=0.1. Fig. 6. View largeDownload slide The state x3 of system (5.1) and the estimated state x^3 of reduced-order observer (5.3) when θ=0.1. Fig. 6. View largeDownload slide The state x3 of system (5.1) and the estimated state x^3 of reduced-order observer (5.3) when θ=0.1. Fig. 7. View largeDownload slide The state x1 of system (5.1) and the estimated state x^1 of adaptive full-order observer (5.4) when θ=1. Fig. 7. View largeDownload slide The state x1 of system (5.1) and the estimated state x^1 of adaptive full-order observer (5.4) when θ=1. Fig. 8. View largeDownload slide The state x2 of system (5.1) and the estimated state x^2 of adaptive full-order observer (5.4) when θ=1. Fig. 8. View largeDownload slide The state x2 of system (5.1) and the estimated state x^2 of adaptive full-order observer (5.4) when θ=1. Fig. 9. View largeDownload slide The state x3 of system (5.1) and the estimated state x^3 of adaptive full-order observer (5.4) when θ=1. Fig. 9. View largeDownload slide The state x3 of system (5.1) and the estimated state x^3 of adaptive full-order observer (5.4) when θ=1. Fig. 10. View largeDownload slide The state x2 of system (5.1) and the estimated state x^2 of reduced-order observer (5.5) when θ=.1 Fig. 10. View largeDownload slide The state x2 of system (5.1) and the estimated state x^2 of reduced-order observer (5.5) when θ=.1 Fig. 11. View largeDownload slide The state x3 of system (5.1) and the estimated state x^3 of reduced-order observer (5.5) when θ=1. Fig. 11. View largeDownload slide The state x3 of system (5.1) and the estimated state x^3 of reduced-order observer (5.5) when θ=1. 6. Conclusions We deal with the observer design problem for unknown Lur'e DI system with one-sided Lipschitz term in this article. Based on some assumptions, we design both full-order observer and reduced-order observer for the system. Finally, the rotor system is used to show the effectiveness of the presented design method. Funding National Natural Science Foundation of China (61403267, 61403268, 21206100), Natural Science Foundation of Jiangsu Province (BK20130322, BK20130331), Natural Science Fund for Colleges and Universities in Jiangsu Province (13KJB510032), China Postdoctoral Science Foundation (2013M530268, 2013M531401) and China Postdoctoral Science special Foundation Funded (2015T80583). References Abbaszadeh M. Marquez H. J. ( 2010 ) Nonlinear observer design for one-sided Lipschitz systems. Proceedings of 2010 American Control Conference, IEEE, Baltimore, USA, pp. 5284–5289 . Acary V. Brogliato B. ( 2008 ) Numerical Methods for Nonsmooth Dynamical Systems, Applications in Mechanics and Electronics . Berlin : Springer . Brogliato B. ( 2003 ) Some perspectives on the analysis and control of complementarity systems . IEEE Trans. Automat. Contr. , 48 , 918 – 935 . Google Scholar CrossRef Search ADS Brogliato B. Heemels W. 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